How to get a Center Point from multiple Vertex Locations?
I will just assume that you're not gonna to consider weight averaging.

The following script show how you could get those coordinates and deal with them, with some comment for very beginner.

import bpy
import numpy as np

# This line select and get the active object from context
ob=bpy.context.object

# Create a Python structure: dict(), with key = vertex_group.name
# and value = vertex_group.index
GL = {g.index: g.name for g in ob.vertex_groups}
verts = {name: [] for name in GL.values()}

for v in ob.data.vertices:
    for g in v.groups:
        verts[GL[g.group]].append(v)
# All the location(co) and index can be found in "verts[name]" list by its attribute using '.'
# eg. verts[G_name][314].co == Vector((1,2,3)) #True
# The "verts" dict's list store the references list point to the data (which is object) they linked

vertex_group_info = {}
for key, value in verts.items():
    # Using numpy array for convenience
    temp = np.array([(v.co[0],v.co[1],v.co[2]) for v in value])
    max = np.max(temp,axis=0) #calculate the max
    min = np.min(temp,axis=0) #calculate the min
    # Average the cooridinates by add them and divide by number of vertices
    center = np.sum(temp, axis=0) / temp.shape[0]
    vertex_group_info[key] = {"max": max,
                               "min": min,
                               "center": center}


What is weighted version?

Let say we got 3 points assign to a group with 1.0, 0.02, 0.01 weight. You might want to calculate the weighted average, which is very closed to the 1st point since 2nd and 3rd show little to no influence to the group. The script above still treat these vertices as same weight if the weight value is not equal to 0 (not assigned).

In the weighted case, you will need to multiply the coordinates by the vertex weight and sum all these weighted coordinates. Than divided it by the sum of vertex weight.